Contents

Idea

General

Given a fiber sequenceF→A→cBF \to A \stackrel{\mathbf{c}}{\to} B of classifying spaces/moduli stacks, hence [c][\mathbf{c}] a universal characteristic class, and given an “AA-structure” in the form of a morphism (cocycle) f:X→Af : X \to A, then a lift f^\hat f through F→AF \to A to an “FF-structure” exists precisely if the induced BB-structure c(f):X→B\mathbf{c}(f) : X \to B is trivializable in BB-cohomology. One says that [c(f)][\mathbf{c}(f)] it is the obstruction to lifting the AA-structure to an FF-structure.

be any other function. We are asking for the obstruction to lift it to a function f^:X→F\hat f : X \to F such that

i∘f^≃f.
i \circ \hat f \simeq f
\,.

This exists precisely if there is an equivalence

ϕ:c∘f→≃ptB
\phi : \mathbf{c}\circ f \stackrel{\simeq}{\to} pt_B

hence if the obstruction class

f*c≔c∘f
f^* \mathbf{c} \coloneqq \mathbf{c} \circ f

is trivial.

Examples

Lift through Postnikov stages

If F→AF \to A in the above is a stage τ≤n+1B→τ≤nB\tau_{\leq n+1} B \to \tau_{\leq n}B in the Postnikov tower of an object BB, then the lifting problem is that of lifting through the Postnikov tower of AA and the universal obstruction class is that which classified τ≤n+1B→τ≤nB\tau_{\leq n+1} B \to \tau_{\leq n}B as a πn+1B\pi_{n+1} B-principal infinity-bundle.

Obstruction to extension

The formal dual of the lift obstruction problem discussed above is the following extension problem:

representing a class [c]∈Hn(BG,A)[\mathbf{c}] \in H^n(\mathbf{B}G, A) in the AA-cohomology of BG\mathbf{B}G. Then given a morphism ϕ:BG→BH\phi : \mathbf{B}G \to \mathbf{B}H we may ask for the obstruction to extending c\mathbf{c} along it.

Now the statement is: if ϕ\phi is a homotopy cofiber, then there is a good obstruction theory to answer this question. Namely in that situation we are looking at a diagram of the form

where the left square is an homotopy pushout. By its universal property, the extension c^\hat {\mathbf{c}} of c\mathbf{c} exists as indicated precisely if the class

[f*c]∈Hn(BQ,A)
[f^* \mathbf{c}] \in H^n(\mathbf{B}Q, A)

is trivial.

One class of examples for this sort of situation is where one considers refined Lie group cohomology on simply connected Lie groups and is asking for ways to push it down to discrete quotients, hence to non-simply connected Lie groups integrating the same Lie algebra. This is often phrased in terms of “multiplicative bundle gerbes” over these Lie groups, but that is just another way of talking about the corresponding cohomology of the smoothmoduli stackBG\mathbf{B}G.

Obstruction to quantization: Quantum anomaly

There are various formalizations of the notion of quantization in physics, or at least various aspects of that formalization. This involves various steps, some of which may have obstructions to being carried out. In physics such an obstruction in the process of quantization is often called a quantum anomaly.